How should I design this LSTM network to perform stock prediction?
I can't say for all cases, but I can certainly give you an example of how an LSTM could be used to predict stock prices.

An LSTM is temporal, meaning you can feed in one input, get an output and the network will remember this interaction (if it's important), which will effect the outcome of future predictions. As such, it is reasonable to feed in all the data for the day (Open, Close etc), obtain an output for the predicted stock price of the following day (or a prediction for all the inputs you feed in, the output can be as large or small as you want), then repeat for as long as you want, each time only feeding in the data for that particular day as the network will remember previous days.

In your example, 60 timesteps would mean doing a forward pass of the LSTM 60 times, for 60 days. Each of these forward passes will produce an output, which you can compare to the next actual stock price for verification, until you reach the current day, where the prediction is for the future.